SATURATION ABSORPTION SPECTROSCOPY ON Cs …...electrons orbiting the nucleus. Some of these elements have very precisely known hyperfine atomic transitions e.g. Rb or Cs and, as

SATURATION ABSORPTION

SPECTROSCOPY ON

Cs ATOMS

Amr Mohamed Ibrahim

Master Thesis

April 2016

Institute of Photonics

University of Eastern Finland

Amr M. Ibrahim Saturation Absorption Spectroscopy on Cs Atoms, 41 pages

University of Eastern Finland

Master’s Degree Programme in Photonics

Supervisors Prof. Yuri Svirko

Prof. Iain Moore

Co-Supervisor Dr. Annika Voss

Abstract

Saturation absorption spectroscopy is a Doppler-free technique by which the hyper-

fine atomic structure of an element such as Rb or Cs can be obtained. When this

technique is used in conjunction with a lock-in amplifier one can get a noise-free

signal that can be used as a feedback for locking a laser frequency against long-term

frequency drifts. In this thesis, the saturation absorption spectroscopy was applied

on Cs atoms in a vapor cell.

The Doppler-broadened absorption spectrum of Cs was measured followed by a

Doppler-free absorption spectrum using saturation absorption spectroscopy. There-

after, the lock-in amplification technique was applied in case of the latter and inves-

tigated at numerous chopping frequencies ranging from 40 Hz to 2 kHz and across

different phases at the frequency of 400 Hz.

Preface

Gaining knowledge lightens our road in life to understand nature and achieve progress.

In this context, I would like to express my deep gratitude to every person who par-

ticipated in educating me. I am indebted to all my professors, researchers and our

coordinator Dr: Noora Heikkila at the Institute of Photonics, University of Eastern

Finland for their teaching and willingness to always support me during the two years

of my master degree. In particular, I would like to thank my supervisor Prof. Yuri

Svirko for his supervision to this work at UEF and for the valuable courses he taught

me with other students and his readiness and patience for answering questions either

during his lectures or in his office. I am very grateful to Prof. Iain Moore at IGISOL,

Jyvaskyla for giving me the opportunity to join his research group and for supervis-

ing my work and teaching me the nuclear physics spectroscopy techniques for one

of which my work will be helpful. For my co-supervisor Dr: Annika Voss, I am very

much thankful for your dedication during supervising my work and instructing me

in the laboratory without which I would not have been be able to complete the work

in due time. Thanks for all the nice people I got to know in Europe and the new

friends I have had who enriched my life experience and their through support made

my life joyful.

Jyvaskyla, the 27th of April 2016 Amr M. Ibrahim

iii

Contents

1 Introduction 1

2 Atomic Spectra and Laser Spectroscopy 4

2.1 Atomic transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.1 Fine structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.2 Hyperfine structure . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.3 Natural linewidth . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Absorption and emission spectroscopy . . . . . . . . . . . . . . . . . . 8

2.2.1 Saturation absorption spectroscopy . . . . . . . . . . . . . . . 10

2.2.2 Cesium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Lock-in amplifier technique . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.1 Typical noise sources . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.2 Lock-in amplifier . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.3 Phase sensitive detection . . . . . . . . . . . . . . . . . . . . . 15

2.3.4 Mathematical description . . . . . . . . . . . . . . . . . . . . 16

3 Experimental setup 18

3.1 Saturation absorption spectroscopy setup . . . . . . . . . . . . . . . . 18

3.2 Lock-in amplifier setup . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3 Data acquisition system and the lock-in amplifier operation . . . . . . 25

3.3.1 LabVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

iv

4 Results and Analysis 27

4.1 Cs spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 Lock-in amplifier implementation . . . . . . . . . . . . . . . . . . . . 28

4.3 Analysis of Cs lock-in spectra . . . . . . . . . . . . . . . . . . . . . . 30

4.3.1 Lock-in amplifier signal dependence on phase . . . . . . . . . 33

4.3.2 Lock-in amplifier signal dependence on frequency . . . . . . . 33

5 Summary and outlook 38

Bibliography 39

v

Chapter I

Introduction

The Doppler-broadened distribution of an atomic absorption spectrum constituted

an obstacle in the past to the identification of the atomic transition lines and mea-

surement of their natural linewidths. Nonetheless, with the advent of tunable lasers

and in turn laser spectroscopy, techniques have been developed to tackle that diffi-

culty. One such technique is saturation absorption spectroscopy. The technique is

essentially a Doppler-free spectroscopy tool such that the hyperfine atomic structure

of an element like Rb or Cs can be readily revealed.

The utilization of the lock-in amplifier in conjunction with saturation absorp-

tion spectroscopy has been applied in numerous applications. For example, the

technique formed the basis for developing high-resolution laser spectroscopy [1], for

which Arthur L. Schawlow shared a Nobel Prize in physics in 1981. Also, it has been

exploited to address the long-term frequency stability of different types of lasers [2].

In addition, the technique paved the way for enabling the laser cooling of atoms and

molecules [3] as well as Bose-Einstein condensate [4].

The significance of lasers is not limited to merely studying the atomic and molec-

ular properties, they are also employed in nuclear physics research. For instance,

they have been used for selecting and enhancing the production of the radioactive

ions produced by accelerators and in the study of their exotic properties. One of the

most often used laser spectroscopy techniques is collinear laser spectroscopy. In this

kind of spectroscopy, a cooled/bunched radioactive ion beam is overlapped with a

co- or counter-propagating laser beam. It allows the study of nuclear ground state

properties such as nuclear charge radii, nuclear spins, electric quadrupole moments,

etc. Continuous wave (CW) laser beams are favored rather than pulsed lasers since

1

their linewidth is significantly narrower. For a recent review of the field, the reader

is referred to [5].

Collinear laser spectroscopy always demands high-resolution laser beams. This

implies that the laser beams should have a long-term stability against frequency

drifts. The requirement of short-term stability of current CW lasers is already sat-

isfied. However, these lasers are only stable over a few hours of use or less while

the collinear laser experiments may last sometimes for a week. The difficulty of

attaining a long-term laser stability for solid state lasers has been the motivation for

the work of this thesis. The aim is to lock a Matisse CW laser at a certain frequency

and to stabilize it against long-term drifts that occur owing to various disturbances

that the laser is subjected to. This goal is pursued by locking the laser frequency

to a well-known hyperfine structure transition of an element such as Rb or Cs. In

this regard, one can utilize the technique of ”saturation absorption spectroscopy”

wherein the hyperfine structure of an element can be readily seen.

Previously at the IGISOL faciltiy, Jyvaskyla [6], saturation absorption spec-

troscopy was applied to Rb for the characterization of Fabry-Perot interferometers

(FPI) [7], [8]. In this thesis, the same technique is applied to Cs in conjunction

with a lock-in amplifier for attaining the goal of long-term laser beam stability. The

importance of involving the lock-in amplifier lies in the reduction of the noise accom-

panied by the detected laser beam signal. Ordinarily, the detected signal is affected

by different noise sources, for example temperature/pressure variations in the laser

laboratory, electronic noise of the used devices and the mechanical vibrations caused

either by the persons working in the laser room or by external equipment such as

vacuum pumps, etc. Ultimately all these sources influence the signal to which the

laser frequency can be locked.

This work focuses on measuring the Cs hyperfine structure of the D2 line at

λ ∼ 852 nm and applies the lock-in amplifier technique to minimize the background

noise associated with the measured Cs spectrum. In this system the laser is chopped

at a particular frequency by an optical chopper wheel. Investigations of various chop-

ping frequencies have been performed and the phase dependence of the lock-in signal

has been studied.

In chapter 2, the theoretical underpinnings of the saturation absorption spec-

troscopy as well as the lock-in amplifier technique are explained. Chapter 3 illus-

trates the experimental setups of both the saturation absorption spectroscopy in

2

addition to the lock-in amplifier. Moreover, this chapter discusses how the data

acquisition system works. The results obtained through this work are shown and

analyzed in chapter 4. A summary of the overall work performed in this thesis and

an outlook is presented in chapter 5.

3

Chapter II

Atomic Spectra and Laser Spectroscopy

The chemical elements in nature and their corresponding isotopes often possess

hyperfine atomic structure due to the coupling between the spin of the nucleus and

the total angular momentum (electron spin and orbital angular momentum) of the

electrons orbiting the nucleus. Some of these elements have very precisely known

hyperfine atomic transitions e.g. Rb or Cs and, as such, they can be exploited for

long-term frequency stabilization of a laser. Saturation absorption spectroscopy is

one of the known techniques by which such a goal is attained, and will be discussed

in this chapter.

2.1 Atomic transitions

In 1900, Max Planck introduced the concept of energy quanta [9] which revolu-

tionized our understanding of nature and was a milestone for numerous advances

within the atomic and subatomic world. Five years later, Albert Einstein used

his idea and conceptualized the quantization of light [10]. Later, light quanta were

named ”photons”. These counter-intuitive ideas led Niels Bohr to propose his atomic

model [11–13] wherein he considered the electrons moving around the atoms in well-

defined orbits (energy levels). Moving from one orbit or energy level to another

requires specific amounts of energy in a quantized manner. If an electron is pro-

moted to an upper energy level, it must absorb a light quantum corresponding to

the energy difference between these two levels. In contrast, if the electron relaxes

to a lower energy level, it emits an energy quanta bearing the difference in energy.

The energy levels and electron movements between them are so-called atomic tran-

sitions. Studying those transitions is the core of atomic spectroscopy. Lasers are an

4

instrumental tool for probing precisely such atomic transitions. An illustration of

an atomic transition can be seen in Fig. 2.1 [14].

Figure 2.1: Example of an atomic transition (adapted), Ref. [14] .

2.1.1 Fine structure

The spectral lines of an atom are not singlets in reality. If we look at the atomic

emission spectrum of any atom with adequately high resolution one can see that

the spectral lines are split into slightly separated lines [15]. The phenomenon is

attributed to the interaction between the electron spin denoted by S and the orbital

angular momentum of that electron around the atom denoted by L. The sum of

electron spin and its angular momentum is denoted by J such that J = S +L. The

electron can be considered as a point-like moving electric charge that has an associ-

ated small magnetic moment. The coupling between the magnetic field field due to

the electron spin’s with the magnetic field due to the orbital angular momentum is

called the spin-orbit interaction. Since the Hydrogen atom has the simplest atomic

structure, its spectrum and every level splitting are shown in Fig. 2.2 [16], as an

example for illustration.

A dimensionless quantity called the fine structure constant denoted by α is used

to characterize the splitting between two adjacent spectral lines. This constant is

calculated from the equation

α =ke2

hc= 7.29735254× 10−3, (2.1)

where k is Coulomb’s constant, e the electron charge, h Planck’s constant and c the

speed of light.

5

Figure 2.2: Hydrogen atom fine structure, Ref. [16].

2.1.2 Hyperfine structure

The spin-orbit interaction is not the only interaction that exists in the atomic system.

There is an additional interaction which should be taken into account in order to

fully understand the atomic spectra. That interaction is between the electron and

the nucleus and further splits the atomic fine structure into several spectral lines.

The interaction is attributed to the coupling between the nuclear magnetic moment

caused by the nuclear spin denoted by I and the magnetic field generated by the

electrons at the site of the nucleus. The coupling sum is denoted by F , where

F = I + J . Returning again to the Hydrogen atom example, Fig. 2.3 presents

Figure 2.3: Fine and hyperfine structures of a Hydrogen atom, Ref. [14]. n

represents the principal quantum number.

6

the previously discussed fine structure and the further splitting representing the

hyperfine structure [14].

2.1.3 Natural linewidth

When an atom is excited by e.g. a light source, it makes a transition from its

atomic ground state to one of its excited states. The period of time in which the

atom resides at the excited state is finite and is the so-called lifetime of the excited

state. After that period, the atom spontaneously decays emiting an energy quanta, a

well-known process is called spontaneous emission. Since the spontaneous emission

is a random process and each excited state has its own lifetime, there will be a

distribution of the emitted light intensity upon relaxation. Furthermore, for each

particular excited state, there are no definite energy levels at which single intensity

lines can be obtained at absorption or emission frequencies. Heisenberg’s uncertainty

Figure 2.4: The relationship between the energy uncertainty and the transi-

tion lifetime, Ref. [17].

principle relates the uncertainty of the energy of each level to the lifetime of the

excited state such that ∆E ≃ ~/τ where ∆E is the uncertainty in energy, ~ is the

Planck’s constant divided by 2π and τ is the transition lifetime. The FWHM is

utilized to characterize the linewidth of the atomic transition and is denoted by Γ,

where Γ = ∆E/h . The spectral distribution characterized by Γ has a Lorentzian line

profile. The uncertainty principle and the line profile of the transition are illustrated

in Fig. 2.4.

7

2.2 Absorption and emission spectroscopy

When an atomic sample within a vapor cell is illuminated by electromagnetic ra-

diation from a broad band light source or tunable laser, as depicted in Fig. 2.5, it

interacts with that radiation. This interaction occurs in an element-dependent way

as specific wavelengths are absorbed while others are emitted. In reality, there is no

monochromatic light. There will always be a spectral line distribution of the light

intensity around a central frequency in case of emission or absorption. For example,

if an atomic transition takes place between two energy bands ∆Ek and ∆Ei and the

corresponding frequencies are ν1 and ν2, the spectral distribution of the light inten-

sity emitted, I(ν), will be centralized around a frequency νo as illustrated in Fig. 2.6.

The light intensity function of the spectral distribution around the central frequency

νo is often called the line profile. As illustrated in Fig. 2.6, it is apparent that ν1

is the minimum frequency corresponding to the transition from the top of the ∆Ek

energy band to the bottom of the ∆Ei energy band. On the other hand, ν2 is the

maximum frequency corresponding to the transition between the bottom of the ∆Ek

energy band to the top of the ∆Ei energy band. The maximum intensity peak of

the line profile Io is at the central frequency νo while ν1 and ν2 have a value of Io/2.

Therefore, the interval δν between ν1 and ν2 is called the full width at half maximum

(FWHM) which is used to define the linewidth of any spectral distribution [17]. The

spectral distribution of Fig. 2.6 represents an emission spectrum whereby the light

intensity is a function of frequency. On the contrary, the absorption spectrum is

illustrated in Fig. 2.7. Absorption and emission spectroscopy are used to study the

energy transitions of atoms, ions or molecules [15].

For the atoms in the vapor cell, each atom has its own velocity distribution

along the propagation direction i.e. the Z-axis. The velocity distribution is described

by a normalized Maxwell-Boltzmann distribution,

P (Vz) =

√

m

2πKBTexp

(

−mV 2

z

2KBT

)

, (2.2)

where Vz is the velocity component along the Z-axis, m is the atom’s mass, KB

is Boltzmann’s constant and T is the gas temperature. If a laser beam is passing

through a gas cell and the frequency of this beam is off-resonance with respect to

a transition between two energy levels in the atom, each atom will see a Doppler-

shifted frequency depending on whether they are propagating in the same or opposite

8

Figure 2.5: Sample irradiation with an electromagnetic field. PD1 detects

emission spectrum represented in Fig. 2.6 and PD2 detects absorption spec-

trum represented in Fig. 2.7.

Figure 2.6: Emission line profile of a spectral distribution (adapted), Ref.[ [17]].

direction to the laser beam [19]. The Doppler-shifted frequency can be described by

ν = νo

(

1±Vz

c

)

(2.3)

where ν is the Doppler-shifted frequency, νo is the resonance frequency of the atoms

in the rest frame, c is the speed of light and Vz is the velocity of the atoms along the

Z-axis. As a result, the atoms possess a Doppler-broadened frequency distribution

of line width

∆νD = νo

√

8KBT ln 2

mc2(2.4)

The profile of the Doppler-broadened frequency distribution is a Gaussian function.

9

Figure 2.7: Absorption spectrum of a sample as a function of relative fre-

quency, Ref. [18].

2.2.1 Saturation absorption spectroscopy

Saturation absorption spectroscopy is a Doppler-free technique that facilitates the

precision measurement of the natural linewidth of atomic transitions. The range

of these natural linewidths is typically a few MHz, to a few tens of MHz evidently

narrower than the normal Doppler-broadened linewidth [18]. In this technique,

essentially three beams are utilized; the ”probe”, ”reference” and ”pump” beams.

All three originate from the same laser source but they are split up and propagate

through a vapor cell, e.g. of Cs. Hence, the common feature of these beams is that

they all have the same frequency. Both the reference and probe beams are identical

in power and propagation direction. In contrast, the pump beam has a higher power

and propagates in the opposite direction. The pump beam must perfectly overlap

the probe beam, as shown in Fig. 2.8. The reference and probe laser beams are

detected by photodiodes.

To fully detect absorption peaks, the spectral range over which these peaks occur

must be scanned. If the pump beam is blocked, the probe and reference beam will be

merely recorded and normal Doppler-broadened absorption spectra will be obtained,

as shown in Fig. 2.9(a). Once the pump beam is unblocked, the absorption spectra

will be changed and a tiny dip, the so-called Lamb dip, is observed as illustrated

in Fig. 2.9(b). In fact, the full width at half maximum (FWHM) of the Lamb dip

is effectively the natural linewidth of the detected atomic transition, and it has a

10

Figure 2.8: Operational principle of the saturation absorption spectroscopy

technique.

Lorentzian profile shape. Hence, the Doppler-broadening effect no longer plays a

role in broadening the absorption peaks at the frequency of the Lamb dip. Thus,

that is the reason why this spectroscopic technique is sometimes called Doppler-free

spectroscopy.

Figure 2.9: Absorption spectrum (a) without and (b) with pump beam, Ref. [18].

When the laser beam frequency is off-resonance, ν 6= νo, the atoms moving either

in the same propagation direction or in the opposite direction with respect to the

laser beam see a red-shifted or blue-shifted frequency, respectively. On the other

hand, when the laser beam frequency is on resonance ν = νo, only atoms that move

perpendicular to the laser beam, namely vz = 0 will experience no Doppler shift.

In turn, these atoms are addressed via the pump and probe beams simultaneously.

Assuming that the atoms have a two-level system, and since the pump beam has a

11

higher power than the probe beam, the pump beam will deplete the ground state

level and promote atoms to the excited state. Therefore, the probe beam will find

a reduced number of atoms in the ground state, hence the Lamb dip is created,

illustrated in Fig. 2.9(b).

2.2.2 Cesium

Cesium with an atomic number Z=55, is one of the most abundant materials on

earth. It has numerous known radioactive isotopes varying in mass number from

neutron-deficient 112Cs up to neutron-rich 151Cs [20]. Of these isotopes, the only

stable one is 133Cs; this feature besides its very precise hyperfine structure, gave

rise to the use as an atomic clock for the definition of the second [21]. In this work

the D2 atomic transition of 133Cs is targeted, thus an infrared laser wavelength is

utilized around 852 nm. In the adapted Table 2.1 [22], more details about the optical

properties of the D2 transition are provided.

Table 2.1

Cesium D2 transition optical properties (adapted), Ref. [22].

Optical property Symbol Value

Frequency ωo 2π · 351.72571850 THz

Transition energy ~ωo 1.454620542 eV

Wavelength (vacuum) λ 852.34727582 nm

Wavelength (air) λair 852.11873 nm

Wavenumber (vacuum) kL/2π 11732.3071049 cm−1

Lifetime τ 30.473 ns

Decay rate/Natural linewidth (FWHM) Γ 32.815 × 106 s−1, 2π · 5.2227 MHz

Given the information provided in the table, a depiction of the relevant energy

levels of the Cs atomic hypefine transitions can be made, such as the one presented

in Fig. 2.10.

12

Figure 2.10: Energy level diagram of the Cs atomic hyperfine transitions.

2.3 Lock-in amplifier technique

Obtaining clear signals during the detection of the absorption spectra of an element

can be challenging. As a matter of a fact, the surrounding environment may sub-

stantially affect any kind of recorded signals. This influence is expressed in a form

of background noise. Consequently, any measurement process is perturbed, and in

some cases if the detected signal is weak it could be overwhelmed by the background

noise.

2.3.1 Typical noise sources

The sources of noise are numerous. For example, the noise can be from the vibrations

of adjacent devices and machines, temperature and/or pressure fluctuations in the

measurement room, or from shot noise in photodiodes because of the conversion of

photons to an electrical current. Moreover, there are other noise sources such as

the vibrations from vacuum pumps and light flickering due to 50 Hz AC. Among

these sources, the 1/f noise or what is called the flicker noise is often the dominant

source of background. This noise appears particularly at low frequencies, and it is

a common characteristic, it exists in electronic noise as well as other perturbative

noises. In general, any measurement could be sensitive to any noise frequencies and

13

the effect of that noise could be reduced if the measurement process is performed at

higher frequencies. Fig. 2.11 shows the spectrum of common noise sources in e.g.

a university laboratory, and the appropriate frequency regions where good signal

measurements can possibly be carried out [23]. As shown in the diagram, at low

frequencies the 1/f characteristic is evident, so the power per cycle as a function of

the frequency is quite high while at higher frequencies is considerably reduced.

Figure 2.11: Noise spectral diagram of common noise sources, Ref. [23].

Power/cycle is equiavalent to the power per defined unit of time.

2.3.2 Lock-in amplifier

There are some devices such as lock-in amplifiers which are able to detect very small

AC signals even at the level of a few nanovolts and buried under much larger noise

signals. Lock-in amplifiers use a simple technique, the so-called phase-sensitive de-

tection method that extracts the desired signal at a specific reference frequency and

phase irrespective to the accompanied frequencies due to the noise [24].

A lock-in amplifier can be considered as a special kind of rectifier. It rectifies or

converts the desired signal from an AC signal to a DC signal whilst it suppresses

other present interfering noise. The output DC signal is then singled out from the

accompanied AC noise signals using a low-pass filter and measured via a DC volt-

meter. In order for the detector of the lock-in amplifier to be able to identify the

14

desired signal among the noise, it should be supplied by a reference signal having the

same frequency and phase as the measured signal [25]. In this sense, the reference

signal is locked to the input signal, which gives the amplifier its name. Usually a

chopper wheel acts as a signal modulator.

For example, the scheme of Fig. 2.12 presents the basic setup of a lock-in ampli-

fier utilized to get rid of the noise associated with the measured laser beam through

a Cs vapor cell. In this experiment, the measured signal is the laser beam pass-

ing through the cell. The role of the chopper wheel is to modulate the signal at a

frequency outside the 1/f noise region shown in Fig. 2.11. The lock-in amplifier re-

ceives two input signals, one from the reference, i.e. the chopper wheel and another

signal which comes from the photodiode. The latter signal essentially expresses the

measured laser beam at the modulated frequency converted to electrical AC current

by the photodiode. This signal actually contains noise due to the effect of the sur-

rounding environment as well as the photodiodes themselves. After being processed

inside the lock-in amplifier, the output is a DC signal.

Figure 2.12: Basic setup of a lock-in amplifier used in spectroscopy

(adapted), Ref. [26].

2.3.3 Phase sensitive detection

Phase sensitive detection (or demodulation) is an operation executed by multiplying

two AC signals having the same frequency, one of which the reference signal and

another the input signal. Fig. 2.13 (a) shows an example to explain the operation.

15

In this case, the input signal is assumed to be a noise-free sinusoidal signal and

the reference signal is an internally generated sinusoidal signal. Both reference

and input signals in this example have no phase difference between each other.

The demodulated output (which is a DC signal) following the multiplication has

a positive mean level and the frequency has been doubled. On the contrary, in

Fig. 2.13 (b), the input signal has a phase delay by 90◦ to the reference signal.

Consequently, the DC output signal has a zero mean value [25]. Typically, a phase

shifter is utilized to eliminate the phase difference between the reference and input

signal. Any other accompanied noise normally has random phases. Therefore, by

means of a low-pass filter the output DC signal will be isolated and then measured.

(a) (b)

Figure 2.13: Phase sensitive detection with (a) 0◦ and (b) 90◦ phase delay

between input and reference signal, Ref. [25].

2.3.4 Mathematical description

The technique of phase sensitive detection is based on simple trigonometric calcu-

lations. The following mathematical description is applied to the case of a sim-

ple sinusoidal signal accompanied with noise. Suppose that the input signal is a

noise-associated sinusoid Vsignal = Vsig sin(ωrt + θsig) + n(t) where ωr = 2πFr is

the angular frequency of the signal at which it must be excited from the exter-

nal reference, Fr is the frequency in hertz, θsig is the signal’s phase, and n(t) is a

time-dependent noise. Another reference signal is generated internally to which the

16

phase-locked loop is locked to the external frequency. The output signal of that loop

is Vref = VL sin(ωLt + θref) where ωL is the angular frequency of the internally gen-

erated signal and θref is the phase-shift between the external and internal reference

signals. Because in the phase sensitive detection the input and reference signals are

multiplied, the product is as follows:

Vpsd = [Vsig sin(ωrt+ θsig) + n(t)].[VL sin(ωLt+ θref)]

= VsigVL sin(ωrt+ θsig) sin(ωLt+ θref) + n(t)VL sin(ωLt+ θref)

= 1/2VsigVL(cos(ωrt+ θsig − ωLt− θref)− cos(ωrt+ θsig − ωLt+ θref))

+ n(t)VL sin(ωLt+ θref)

= 1/2VsigVL(cos([(ωr − ωL)t + (θsig − θref)])− cos([(ωr + ωL)t+ (θsig + θref)]))

+ n(t)VL sin(ωLt+ θref). (2.5)

The output of Vpsd multiplication has three AC components, one at the difference

frequency between the reference and input signal (ωr − ωL), another at the sum

frequency (ωr + ωL) as well as the AC noise n(t) sin(ωLt + θref). When the three

components pass through the low-pass filter, they will be washed out and nothing

will be left. However, if ωr = ωL, the difference frequency is converted to DC

current which can pass through the low-pass filter while the other two components

will remain AC but at double the frequency, in other words the ’second harmonic’.

Therefore, the low-pass filter will prevent the higher harmonics from going through,

allowing solely 1/2VsigVLcosθ to be subsequently amplified by the DC amplifier and

measured, where θ = θsig − θref. This phase difference can be eliminated, namely

θ = 0 in the lock-in amplifier such that the output signal becomes 1/2VsigVL.

17

Chapter III

Experimental setup

The experimental setup chapter encompasses three sections: the saturation absorp-

tion spectroscopy setup, the lock-in amplifier and the data acquisition system. The

first section of this chapter provides a close up of the saturation absorption spec-

troscopy setup and the function of each component involved in this spectroscopic

technique. In the second section, the function of the lock-in amplifier interior parts

will be discussed. It is worth noting that the lock-in amplifier operates in conjunc-

tion with the saturation absorption technique, namely it takes the detected signal

of the saturation absorption and separates the associated noise from the signal. The

last section illustrates how all the data are acquisitioned and processed through

the entire system ending up primarily displayed and then recorded by LabVIEW

program.

3.1 Saturation absorption spectroscopy setup

The saturation absorption spectroscopy technique is often implemented on certain

elements possessing well-known hyperfine structures such as Rb or Cs [27], [28], [29]

in order to lock a laser against long-term frequency drifts [30], [31], [32]. In this

thesis, a vapor cell filled with 133Cs is used for the same purpose.

The laser system and the optical table having the saturation absorption experi-

ment is shown in Fig. 3.1. Laser light from a Millenia pump laser at wavelength 532

nm and maximum output power 6 W, is used as a pump beam for the CW Ti:Sa laser

cavity (Matisse TS, Sirah). The emitted Ti:Sa light at ∼ 852 nm strikes the first

mirror M1 and is then reflected onto a second mirror M2. To avoid back reflection

into the laser, an optical isolator is involved which delivers the laser beam to a λ/2

18

wave plate retarder that changes the polarization direction of the transmitted laser

beam. The laser beam passes through a polarizing beam splitter cube (PBSC) which

splits the beam into two paths, one is coupled via an optical fiber to a wavemeter

to measure the wavelength and the other one is sent to a second optical bench with

the saturation absorption setup. On the optical bench the laser beam first passes

through a beam splitter that splits the laser into two beams. One beam is reflected

by a mirror M3 through a lens to focus the laser, which before being reflected by

mirror M4. Since in the saturation absorption technique three beams are required:

pump, probe and reference beam, the laser beam after being reflected via M4 is split

into three beams by means of a thick beam splitter (12 mm) as indicated in the

figure. The highest in power is the pump beam while the reference and probe beams

are almost identical and have lower power. The pump beam goes straight through

the beam splitter to mirror M5 and then passes through a telescope, expanding the

beam diameter. Thereafter, the same beam hits another mirror M6 and then hits a

D-shaped mirror which forwards it towards the Cs vapor cell. That implies the pump

beam is not detected by the two photodiodes. In contrast, the reference and probe

beams follow a different path where upon splitting they immediately pass through

the Cs cell, over the D-shaped mirror and are detected by the two photodiodes. The

Cs cell is a transparent vapor cell of Cs atoms at low density with tilted side edges

of 11◦ to avert back reflections, as shown in Fig. 3.1.

The key instrument in detecting the absorption spectra of Cs in the setup are

the two photodiodes that detect the probe and reference beams. The importance of

these is because they essentially convert the received photons into electrons which

can be then measured as a voltage of an electrical current. Moreover, the two pho-

todiodes are housed in one box with a circuit providing a probe beam signal, a

reference beam signal as well as a difference signal between the probe beam and

reference beam. The reference signal goes to the lock-in amplifier. Initially, the

received optical signal (photons) is converted by means of the two photodiode junc-

tions to an electrical signal (current) that has a small magnitude, hence subsequently

those signals are amplified via the amplifiers within the photodiode box. As seen in

Fig. 3.2, the photodiode box is supplied by 12 V from a battery, and it has three

terminals BNC1, BNC2 and BNC3. The function of BNC1 and BNC2 is to send the

voltage signals of the reference and probe beams to a data acquisition box, to be in

19

Figure 3.1: Overall laser setup for the laser saturation absorption of Cs.

turn recorded by a LabVIEW program. The third terminal BNC3 is dedicated for

delivering the difference signal to the lock-in amplifier.

In the beginning the voltage produced by the two photodiodes owing to the

conversion of photons into electrons was limited to be within the range from 2-10

Volts. If the laser power to the Cs cell is high (≈ 60 µW), the transmitted power

passing out of the cell would saturate the photodiodes, namely the electrical voltage

generated by the photodiodes is more than 10 Volts. If this happens, the electrical

signal (representing the spectrum) is displayed as a straight line all over the scanned

spectral range. Furthermore, the photodiode box interestingly had an offset at 2

V resulting from an earlier use. The ”cut-off” indicating the level of the offset can

be seen in Fig. 3.3. Many systematic tests were performed to try to optimize the

detection. Fig 3.4 indicates the effect of using neutral density filters in front of the

photodiodes in order to reduce the effect of saturation. Eventually, following a num-

ber of adjustments, the sensitivity of the two photodiodes is such that they operate

20

Figure 3.2: Simplified circuit diagram of the photodiode box, Ref. M.

Puskala, 2016, private communication.

within a 0-10 V range.

3.2 Lock-in amplifier setup

The lock-in amplifier, as introduced in Chapter 2, is a device that enables the mea-

surement and detection of very small AC signals within a substantially larger back-

ground noise. One of the lock-in amplifier components is the low-pass band filter

having a tunable narrow frequency bandwidth to suit the measured signal. The

filter allows measurement of the wanted signal by rejecting most of the unwanted

noise. When a lock-in amplifier is utilized in applications, it typically requires a

center frequency of 10 kHz and a bandwidth of 0.01 Hz. Also, the lock-in amplifier

amplifies the input signal after being converted to a DC signal.

The operation of a lock-in amplifier is as seen schematically in Fig. 3.5. Two

signals are provided to the lock-in. One is the reference signal by which the lock-in

recognizes the frequency of the signal of interest in the input signal. The structure of

the lock-in amplifier involves an internal circuit called the phase-lock loop (PLL) by

which the device tracks the input signal frequency. This circuit may be phase shifted

such that its output is cos(ωrt + θref), where ωr is the angular reference frequency

and θref is the phase shift between the external and internal reference signals. The

second signal is the input which is supplied by the experiment and is amplified by

21

Figure 3.3: Standard absorption spectroscopy of Cs with a Cut-off on the

the highest absorption peak.

a high gain AC coupled differential amplifier. The next procedure is that both the

input and PLL reference signal are multiplied via a mixer. As a result, this process

modulates each frequency contained in the input signal ωs by the reference signal

frequency ωr. So, the product has two frequency components, the sum frequency

ωs + ωr and difference frequency ωs − ωr. The low-pass filter then attenuates the

sum frequency but not the difference frequency. The mathematical basis of that

attenuation was explained in Chapter 2. The difference frequency component which

is a DC signal is then amplified and delivered to the data acquisition box that is

connected to a computer.

The lock-in amplifier model used in our setup is SR510 (Stanford Research Sys-

tems, Inc.). Technical specifications and other features of this model are explained

in details in Ref. [33]. The best output signal is attained when the input signal

and reference signal are in phase. This means the phase difference between them

must be zero, based on the mathematical equation of the lock-in amplifier operation.

22

Figure 3.4: Standard absorption spectroscopy of Cs and initial attempts

to optimize the detection on the photodiodes. Note: the difference in refer-

ence+probe signals is due to the signal difference in power between the beams.

This can be achieved by maximizing the output signal using the PHASE keys on the

lock-in amplifier. Alternatively, it could be done using the same PHASE keys, by

setting a phase value provided that the signal is zeroed such that the signal output

is zero. That makes the phase of the reference oscillator 90◦ with respect to the

input signal. Thereafter, by the 90◦ phase keys it should be adjusted to maximize

the output signal. The adjustment this way is rather more sensitive.

The output signal of the lock-in amplifier goes to a data acquisition box which

is connected to a computer such that the data are displayed and recorded by a Lab-

VIEW program.

In order for the lock-in amplifier to perform its function, an optical chopper

wheel must be used. Basically, the optical chopper is a wheel rotating around its

axis at a rotational rate controlled by a chopper controller. The chopper model

SR540 (Stanford Research Systems, Inc.) is used in this work; further information

about the technical specifications is found in Ref. [34]. The function of an optical

23

Figure 3.5: Lock-in amplifier diagram showing the internal process of the

lock-in amplifier operation [adapted], Ref. [33].

chopper wheel is to provide the lock-in amplifier a reference signal at specific fre-

quency by which it can track the desired frequency of the input signal of interest in

the measurement, that is in our case a laser beam signal. This is done by chopping

(or modulating) the pump laser beam at certain frequency.

The chopper wheel of the model SR540 has two types of blades distinct in terms

of the number of slots. In Fig. 3.6, the wheel on the right, has a so-called 6/5 slot

while on the left a 30/25 slot. The number of the blade slots determines the fre-

quency range of operation. For example, the wheel blade which has 30 slots operates

from 400 Hz to 3.7 kHz and the blade of 6 slots could operate from 4 Hz to 400 Hz.

In practice, the safe frequency range for long duration operation of the 30 slot blade

is from 400 Hz to 2 kHz.

Figure 3.6: 30/25 slot and 6/5 slot chopper blade, Ref. [34].

24

3.3 Data acquisition system and the lock-in amplifier oper-

ation

In the following we discuss how the overall data acquisition system works. The

system begins from the photodiode box which provides the aforementioned hardware

difference of the probe and reference signal through its BNC connector where that

signal is considered the input signal of the lock-in amplifier. Since the output voltage

of the photodiode box is a few volts and in turn may overload the lock-in circuit,

a voltage divider of a ratio 1:50000 is used in order to divide the voltage coming

from that box. Thus, the voltage divider input is connected to the photodiode box

BNC connector while the output of the voltage divider is plugged into the lock-in

amplifier itself at terminal A, as indicated in the diagram of Fig. 3.7.

Figure 3.7: Data acquisition system diagram and lock-in amplifier operation.

The external reference signal is supplied from the chopper controller directly to

the lock-in amplifier. The noise-free output of the lock-in amplifier is supplied to

the ’Data acquisition box’ to be presented on a computer screen via the LabVIEW

25

program.

3.3.1 LabVIEW

For displaying and recording the saturation absorption spectra of the element under

study a LabVIEW program is available. Fig. 2.9 presents the user interface of the

program.

Figure 3.8: Screenshot of LabVIEW program layout.

As seen in the screenshot, the program layout consists of four sections, each

serves a certain task. The upper left screen shows the spectrum instantly taken via

the probe and reference beams where the x-axis is the number of FPI scans and the

y-axis is the photodiode voltage. The upper right screen is dedicated for showing

the difference between the pump/probe spectrum and the reference spectrum. In

fact, there are two sorts of that difference, one calculated by means of the LabVIEW

program labeled as ’A-B’ and the other one is provided by the photodiode circuit as

such, labeled as ’Hardware difference’. Below those two panels, there is additional

panel to display the scan of fringes from a Fabry-Perot interferometer. The far

most right of the LabVIEW interface is dedicated for vital information such as the

wavelength as detected by the wavemeter at the frequency of the laser, the exposure

time of the wavemeter, the rate at which the readout is sampled, etc. Equally

important are the options regarding data saving, file name, folder path and so on.

26

Chapter IV

Results and Analysis

Over the course of the thesis work, many scans of the laser frequency across the Cs

hyperfine structure have been performed, and a lot of data have been obtained. This

includes saturation absorption spectroscopy as a standalone technique followed by

the investigation of the lock-in amplifier effect on the saturated Cs peaks. The results

of these investigations will be presented. Furthermore, the chopping frequency as

well as the phase dependence of the lock-in method will be analyzed and discussed.

4.1 Cs spectrum

The first acquired data was the standard Cs absorption spectrum. This was obtained

by allowing only a single laser beam to pass through the Cs vapor cell (either probe

or reference), illustrated in Fig. 2.5. The results are shown in Fig. 4.1. The wave-

length of the laser beam was scanned within an approximate range from 852,325 nm

to 852,363 nm.

Next, the pump beam is overlapped with the probe beam in order to saturate

the ground state transition of Cs atoms, hence the Lamb dips appear. Since the

LabVIEW program subtracts the reference beam signal from the probe beam, what

is indicated in Fig. 4.2 is the difference signal calculated by the program showing

merely the Lamb dips that represent the Doppler-free spectra of Cs atomic transi-

tions.

Two clear negative signals can be seen in the data of Fig. 4.2. The broader

peak on the right is caused by residual Doppler broadening. As illustrated in the

general scheme of Fig. 3.1, a thick beamsplitter is used to produce the probe and

reference beams, thus the two beams are not perfectly matched in power. During

27

Figure 4.1: Normal Cs absorption spectrum.

substitution, this results in a non-zero offset (hence the background voltage on Fig.

4.2 is ∼ 0.2 V off-resonance) and possible small Doppler contribution.

The peak on the left however is sharp. This is of current interest and is under

further investigation.

4.2 Lock-in amplifier implementation

From the previous figures one can clearly see the influence of the background noise

on the obtained Cs spectra. This is most obvious as one scans the wavelength off the

resonance positions. In order to minimize the noise, the same saturated absorption

scans were taken simultaneously with operation of the lock-in amplifier. The pump

beam was chopped by the chopper wheel at a certain frequency. Arbitrarily, chop-

ping at 400 Hz was selected in order to investigate how the lock-in amplifier affects

the spectrum. The full spectrum of Cs was measured with a phase difference of 0◦

28

Figure 4.2: Saturation absorption spectrum of Cs showing Doppler-free tran-

sitions. The reference signal has been subtracted from the probe signal

between the input signal (provided by the photodiode box) and the reference signal

(which comes from the optical chopper wheel). Referring to the mathematical de-

scription of the operational principle of the lock-in amplifier in chapter 2, the signal

measured at 0◦ phase difference should give the optimal result. In addition to 0◦

phase difference, other phase differences were investigated, namely 90◦ and 180◦ at

the same chopping frequency. According to the relationship Vout = 1/2VsigVL cos θ

deduced from Eq.(2.5), the former should give a zero signal (in reality, it gives a

very small signal above or below zero), while for the latter, it should give the maxi-

mum voltage with negative sign. Fig. 4.3 presents all the three investigated phase

differences. As noted in the figure, the signal at 90◦ phase difference is barely seen.

Remarkably, the background noise is almost flat and approximately zero, either at

29

0◦ or at 180◦ phase difference. It should also be noted that the residual Doppler

effect seen in Fig. 4.2 has now been removed. The sharp negative peak seen in the

left of the main Cs structure is still visible. This suggests that the peak is not an

artifact of Doppler broadening and is real.

The obtained lock-in signal at the 90◦ phase difference is better manifested in

Figure 4.3: Full Cs spectrum at 400 Hz with 0◦, 90◦ and 180◦phase difference.

Fig. 4.4. It is likely that at 90◦ the phase was not matched perfectly, thus a small

residual signal is seen, which is noted by the difference in the scale of voltage.

4.3 Analysis of Cs lock-in spectra

Saturation absorption spectroscopy is a Doppler-free technique, thus one does not

expect a Gaussian constribution to the spectral line which would naturally arose in

30

Figure 4.4: Full Cs spectrum at 400 Hz with 90◦ phase difference.

a Doppler-broadened spectrum. In order to look into more detail of the effect of the

lock-in amplifier, we focus on one of the Cs hyperfine multiplets. Fig. 4.5 shows

an example of the lock-in spectra of the upper multiplet. There are 5 clear peaks,

however according to the selection rules, only 3 peaks should be visible (see Fig.

3.4). The peaks have been labeled according to their relevant hyperfine transitions,

namely FL = 4 to Fu = 5, FL = 4 to Fu = 4 and FL = 4 to Fu = 3. Two

additional peaks are seen, labelled ”C.O.”. There are so-called ”cross-over” peaks,

which appear in atoms with more than two hyperfine levels (e.g. Cs) and occur

when two transitions are within a single Doppler-broadened feature and share a

common ground state. An additional peak, the ”cross-over” peak appears in the

middle between the two contributing hyperfine transitions. These extra peaks in

saturated absorption spectra are the result of moving atom experiencing the pump

and probe beams resonantly with two separate transitions. They are therefore a

31

direct consequence of the counter-propagating (pump+probe) beams and are often

stronger than the ”normal” saturated absorption peaks.

In this work we have chosen to fit the strongest peak in the multiplet, which

happens to be a cross-over peak. The fit is performed with a Lorentzian function

(assuming no Doppler contribution);

Figure 4.5: Peak fitting by a Lorentzian function.

y = y0 +2A

π

w

4(x− xc)2 + w2, (4.1)

where the parameters involved in this equation are: y0 is the offset between the

zero line on the y-axis and the peak flat background, A is the area under the fitted

peak, w is the FWHM of the peak and xc is the wavelength value at the center of

the peak. The shown fitted peak here has a FWHM = 7, 9655 × 10−5 nm. This

width, ∆λ, can be converted into frequency units. When doing so, the resulting

32

width, ∆ν = 32.9 MHz. The natural linewidth of the D2 transition in Cs according

to Table 2.1 is 32.8 MHz. Thus, we have achieved almost perfect agreement with

the natural linewidth.

Throughout the subsequent analysis, we focus on the strongest absorption peak

of Fig. 4.5 and use the Lorentzian function to extract the peak height (the lock-in

amplifier signal measured in volts). This is used in a study of the frequency and

phase dependence.

4.3.1 Lock-in amplifier signal dependence on phase

The lock-in amplifier signals across different phase angles, each separated by 45◦,

were obtained. Referring to Eq. (2.5) in chapter 2, the lock-in amplifier output is

expected to have a sinusoidal waveform as a function of of phase. Therefore, the

purpose of taking these measurements was to verify the periodicity of the operation

of the lock-in amplifier. Evidently, as seen in Fig. 4.6, the data were excellently

reproduced with a sine function, Eq. (4.2).

y = y0 + A sin(π(x− xc)

w). (4.2)

The parameters of the fit indicate a centroid of xc = −88(5) degrees, a width, w, of

178(3) degrees and the amplitude A = 8.8(2) volts.

4.3.2 Lock-in amplifier signal dependence on frequency

Identifying suitable frequencies for operation of the chopper wheel motivated the

investigation of the lock-in amplifier signal dependence on the frequency. The fre-

quency range that was investigated is from 40 Hz to 2 kHz. In principle, the wheel

can chop up to 3 kHz. Nevertheless, it is not recommended to exceed 2 kHz for

prolonged operation durations.

In the initial work, the photodiode box was supplied by a 10 V battery. After

completing all frequency measurements, a trend was seen: as the frequency increases

the lock-in signal decreases, illustrated in Fig. 4.7. To interpret that trend a number

of systematic checks were made to see if any parameter was drifting. Interestingly,

when the voltage across the battery terminal was measured it showed a value of only

1 V. Hence, the battery was replaced by a power supply with the output voltage set

at 10 V. The measurements were then repeated.

33

-50 0 50 100 150 200 250 300 350 400-10

-5

0

5

10

data fit

Lock

-in s

igna

l (V)

Phase (degrees)

Figure 4.6: Lock-in signal dependence on phase.

Further caution was taken to ensure that the trend was due to the battery effect

therefore the measurements within the frequency range 400 Hz - 2 kHz performed

either with the battery or power supply, have been normalized and compared, Fig.

4.8. It is noticed that in the measurements of the battery, the normalized lock-in

signal decreases almost linearly with the increase in the frequency.

The overall frequency range obtained using the power supply is in Fig. 4.9. The

figure shows fluctuation in the lock-in amplifier signal at low frequencies, particularly

from 40 Hz to 150 Hz. Interestingly, the lock-in amplifier performance was tested at

50 Hz, equivalent to the frequency of AC power in Europe. A high level of noise was

expected, especially as power lines are well-known noise sources as indicated in Fig.

2.11 of chapter 2. Indeed, the minimum lock-in signal was found at that frequency.

One can also see a second clear minimum at 100 Hz, twice the frequency of 50 Hz.

The data suggests that a preferable frequency to use in connection with the optical

34

Figure 4.7: Lock-in signal as a function of frequency when the battery was

used to provide input voltage to the photodiode box.

chopper wheel would be between 500 Hz to 1500 Hz, where the signal-to-noise ratio

is good. This fits well with the ”quiet region” illustrated in Fig. 2.11.

35

Figure 4.8: Normalized lock-in signal obtained with the battery and the

power supply

36

Figure 4.9: Lock-in signal as a function of frequency. Data taken from 40 Hz

to 2 kHz. Power supply was used.

37

Chapter V

Summary and outlook

Throughout this thesis the saturation absorption spectroscopy on Cs, in conjunction

with the lock-in amplifier technique was reported. At first, the full range of the

absorption spectrum of Cs was measured without the lock-in amplifier. As a result,

an evident effect of the background noise was observed. Thereafter, the lock-in

amplifier was included and a full range of the absorption spectrum at 0◦, 90◦ and 180◦

phase difference between the signal and reference was investigated. In agreement

with the theory of the lock-in amplification, the optimum spectrum was obtained

at 0◦. Furthermore, additonal phase data between 0◦ and 360◦ showed an almost

perfect agreement with the expected cosine function. The effect of the modulation

frequency of the signal was studied using an optical chopper wheel from 40 Hz to 2

kHz. A continuous drop in the lock-in signal intensity was observed from 400 Hz.

Within the investigated frequency range, a quiet operational region from 500 Hz to

1500 Hz has been identified. A noise contribution from the power line at 50 Hz as

well as at 100 Hz was observed, that considerably reduced the lock-in signal.

Similar studies can be performed on Rb in order to cross-check the Cs data.

In the future we will use either Rb or Cs lock-in spectra and then differentiate to

get an error signal which we can feed into the Matisse laser to provide a long-term

stable lock to an ”absolute” frequency standard. This lock may then be compared,

via a Fabry-Perot interferometer, to a commercial stabilized HeNe laser to check for

long-term frequency drifts in the HeNe. The objective of such investigations is to

lock the Matisse laser to a long-term reference such that questions in collinear laser

spectroscopy may be addressed.

38

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41